Saturation Optics

ABSTRACT

An imaging system includes a detector for receiving electromagnetic energy and generating sampled data in accordance with the electromagnetic energy so received. The detector is characterized by a threshold point such that the sampled data is in one of two states: i) below threshold, when the intensity of the electromagnetic energy so received is less than the threshold point; and ii) above threshold, when the intensity of the electromagnetic energy is greater than the threshold point. The imaging system also includes saturation optics for providing a characteristic of the sampled data, wherein the characteristic of the sampled data when below threshold is different from the characteristic of the sampled data when above threshold.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. provisional patent applicationSer. No. 60/802,724, filed May 23, 2006, entitled SATURATION OPTICS;U.S. provisional patent application Ser. No. 60/808,790, filed May 26,2006, entitled SATURATION OPTICS; and PCT patent application serialnumber PCT/U.S.07/09347, entitled ARRAYED IMAGING SYSTEMS AND ASSOCIATEDMETHODS, filed on Apr. 17, 2007, all of which applications areincorporated herein by reference in their entireties.

U.S. patent application Ser. No. 10/376,924, filed Feb. 27, 2003 andentitled OPTIMIZED IMAGE PROCESSING FOR WAVEFRONT CODED IMAGING SYSTEMS,is expressly incorporated herein by reference in its entirety.

BACKGROUND

In typical imaging situations, an imaging system is designed for bestperformance and production of sufficient quality images for averageexpected lighting conditions. Images may be collected from a largevariety of conditions and locales; indoor scenes with artificiallighting, such as incandescent or fluorescent lighting, may varysignificantly from outdoor scenes under bright sunlight.

Composite lighting conditions, mixing indoor and outdoor lightingcharacteristics, are also possible. For example, the imaging system maybe located inside of a building while imaging a scene that includespoorly illuminated dark-colored objects within the building as well asbrightly sunlit light-colored objects outside of the building. Such acombined indoor and outdoor scene may require imaging of a set ofobjects that range in intensity over multiple orders of magnitude.

Intensity of a scene or an object may be formally characterized bymeasuring the luminance of specific objects within the scene. Luminanceis defined as the number of candelas per square meter (“cd/m²”). Forinstance, a dark wooden surface may have a luminance of less than 1cd/m², a light colored wall may have a luminance of approximately 10cd/m², a concrete parking lot may have a luminance of approximately1,000 cd/m² and the sky may have a luminance of greater than 10,000Cd/m².

SUMMARY

In one embodiment, an imaging system for imaging electromagnetic energyis disclosed. The imaging system includes a detector for receiving theelectromagnetic energy and generating sampled data in accordance withthe electromagnetic energy so received. The detector is characterized bya threshold point such that the sampled data is in one of two states: i)below threshold, wherein the intensity of the electromagnetic energy soreceived is less than the threshold point; and ii) above threshold,wherein the intensity of the electromagnetic energy is greater than thethreshold point. The imaging system also includes saturation optics forproviding a characteristic of the sampled data, wherein thecharacteristic of the sampled data below threshold is different from thecharacteristic of the sampled data above threshold. In a furtherembodiment, the saturation optics includes imaging optics for directingthe electromagnetic energy toward the detector and phase modifyingoptics for modifying a wavefront of the electromagnetic energy. Inanother embodiment, the saturation optics include an arrangement ofsegments providing multi-fold symmetry.

In one embodiment, phase modifying optics for use in an imaging systeminclude a constant profile path surface including a plurality ofsegments, each of the plurality of segments including a surface sagdefined by a one dimensional function along a straight lineperpendicular to a radial vector from a center of the phase modifyingoptics.

In one embodiment, a method for designing a pupil function for use in animaging system is disclosed. The method includes selecting a pupilfunction and calculating a sampled PSF, taking into accountcharacteristics of the imaging system and the pupil function. The methodfurther includes evaluating the sampled PSF in accordance with aselected metric and, if the sampled PSF does not conform within theselected metric, then modifying the pupil function, using a set ofparameter modifications, and repeating the evaluating and modifying ofthe pupil function until the sampled PSF conforms within the selectedmetric.

BRIEF DESCRIPTION OF THE DRAWINGS

It is noted that, for purposes of illustrative clarity, certain elementsin the drawings may not be drawn to scale. It is also noted that, forpurposes of illustrative clarity and reproduction compatibility, certainimages may be, without loss of generality, simplified and contrastenhanced.

FIG. 1 is a schematic diagram of an imaging system including saturationoptics, shown here to illustrate a response of the imaging system tonon-saturating illumination, according to an embodiment.

FIG. 2 is a schematic diagram of the imaging system including saturationoptics, shown here to illustrate a response of the imaging system tosaturating illumination, according to an embodiment.

FIG. 3 is a flowchart for a process of designing a pupil function forsaturation optics, according to an embodiment.

FIG. 4 is an illustration of a circular pupil function with spatiallyuniform segmentation, according to an embodiment.

FIG. 5 is an illustration of a circular pupil function with spatiallynon-uniform segmentation, according to an embodiment.

FIG. 6 is an illustration of a circular pupil function with spatiallyuniform segmentation of eight sectors, according to an embodiment.

FIG. 7 is an illustration of details of one sector of the circular pupilfunction shown in FIG. 6.

FIG. 8 shows a plot of an exemplary polynomial for defining a portion ofa pupil function, according to an embodiment.

FIG. 9 shows a three-dimensional (“3D”) mesh plot of an exemplary pupilfunction, according to an embodiment.

FIG. 10 shows a 3D mesh plot of an exemplary smoothing function.

FIG. 11 shows a 3D mesh plot of a modified pupil function combining theexemplary pupil function of FIG. 9 with the exemplary smoothing functionof FIG. 10, according to an embodiment.

FIGS. 12-15 show images of sampled point spread functions (“PSFs”)obtained with an imaging system including the pupil function of FIG. 9,according to an embodiment.

FIGS. 16-19 show images of sampled PSFs obtained with an imaging systemincluding the modified pupil function of FIG. 11, according to anembodiment.

FIG. 20 shows the 3D mesh plot of FIG. 9, repeated here for convenience.

FIG. 21 shows a 3D mesh plot of an exemplary outer region maskingfunction, according to an embodiment.

FIG. 22 shows a 3D mesh plot of an exemplary inner region maskingfunction, according to an embodiment.

FIG. 23 shows a 3D mesh plot of a selected inner region of the pupilfunction of FIG. 9 resulting from point-by-point multiplication of thepupil function of FIG. 9 with the outer region masking function shown inFIG. 21, according to an embodiment.

FIG. 24 shows a 3D mesh plot of a selected outer region of the pupilfunction of FIG. 9 resulting from point-by-point multiplication of thepupil function of FIG. 9 with the inner region masking function shown inFIG. 22, according to an embodiment.

FIG. 25 shows a plot of a saturated sampled PSF obtained with an imagingsystem including the selected inner region of the pupil function asshown in FIG. 23.

FIG. 26 shows a plot of a saturated sampled PSF obtained with an imagingsystem including with the selected outer region of the pupil function asshown in FIG. 24.

FIG. 27 shows the 3D mesh plot of FIG. 11, repeated here forconvenience.

FIG. 28 shows the 3D mesh plot of FIG. 21, repeated here forconvenience.

FIG. 29 shows the 3D mesh plot of FIG. 22, repeated here forconvenience.

FIG. 30 shows a 3D mesh plot of a selected inner region resulting frompoint-by-point multiplication of the pupil function of FIG. 11 with theouter region masking function of FIG. 21, according to an embodiment.

FIG. 31 shows a 3D mesh plot of a selected outer region resulting frompoint-by-point multiplication of the pupil function of FIG. 11 with theinner region masking function of FIG. 22, according to an embodiment.

FIG. 32 shows a plot of a saturated sampled PSF obtained with an imagingsystem including the selected inner region of the pupil function asshown in FIG. 30.

FIG. 33 shows a plot of a saturated sampled PSF obtained with an imagingsystem including the selected outer region of the pupil function asshown in FIG. 31.

FIG. 34 shows a contour plot of a cosine phase pupil function, accordingto an embodiment.

FIG. 35 shows a grayscale image of a random phase pupil function,according to an embodiment.

FIG. 36 shows a grayscale image of the sum of the cosine phase andrandom phase pupil functions respectively shown in FIGS. 34 and 35.

FIG. 37 shows a grayscale image of an unsaturated sampled PSF obtainedwith an imaging system including the pupil function of FIG. 34.

FIG. 38 shows a grayscale image of a saturated sampled PSF obtained withan imaging system including the pupil function of FIG. 34.

FIG. 39 shows a grayscale image of an unsaturated sampled PSF obtainedwith an imaging system including the pupil function of FIG. 36.

FIG. 40 shows a grayscale image of a saturated sampled PSF obtained withan imaging system including the pupil function of FIG. 36.

DETAILED DESCRIPTION OF THE DRAWINGS

As lighting conditions vary, the performance of an imaging system isaffected; therefore we now disclose, among other features, design andimplementation of imaging systems that function in a certain manner whenthe illumination intensity of the electromagnetic energy is sufficientlyhigh to saturate the detector. Two examples of high and low illuminationintensity conditions are: 1) high illumination that saturates a detectorand low illumination that does not saturate a detector; and 2) highillumination that does not quite saturate the detector but the low andhigh illumination intensities are sufficiently different as to bediscernable in the presence of noise. The second definition is forexample useful in connection with binary or thresholded digital images;that is, when considering relative illumination intensities that arebelow a saturation point of the detector, a distinguishing level betweenlow and high illumination intensities may be defined as a value at whichan appropriate thresholding may be determined.

For example, for an eight-bit detector providing electronic data, a lowillumination intensity may mean unsaturated 0 to 254 counts and a highillumination intensity may mean 255 or more counts; that is, for theexemplary eight-bit detector, the value of 255 counts may be denoted asthe saturation point of the detector. Alternatively, for the sameeight-bit detector, illumination levels corresponding to 10 and 100counts may be considered low and high respectively in the presence of anoise level that is less than 90 counts such that the difference inthese illumination levels is discernable.

Saturation of a detector occurs, for example, when the dynamic range ofan imaged scene is greater than the dynamic range of the detector. Forthe exemplary eight-bit detector, for example, a dark wooden surface(less than 1 cd/m²) and a light colored wall (˜10 cd/m²) may notsaturate the detector, whereas a parking lot (˜1,000 cd/m²) and sky(greater than 10,000 cd/m²) may saturate the detector. Theaforementioned object intensities assume that the exemplary detector hasa saturation point at a luminance of, for example, 20 cd/m². Exemplarysaturated sampled PSFs described herein are associated with a saturationlevel. For example, sampled PSF 2500 of FIG. 25 is noted as being 50×saturated. Therefore, PSF 2500 may be associated with an intensity of1000 cd/m² incident upon the detector.

Other than for normal imaging, saturation of a detector may occur inartistic imaging situations or to probe the response of the detector oroptics. Artistic imaging may include the use of structured illuminationsources, specialized flash lighting and/or extended exposure. Probingmay use high intensity incoherent or coherent sources. Probing may beused to determine, for instance, the identification of the detector orfor detection of the detector in military applications. Herein, imagingoptics specially designed for high dynamic range intensity imagingconditions are referred to as “saturation optics.” In other words, animaging system including saturation optics is one designed such that animage formed by the imaging system of an object may be tailored as afunction of, for example, the illumination intensity of the object. Thesaturation optics includes a specially-designed pupil function such thatthe imaging system including the saturation optics behaves in apredetermined way under saturated imaging conditions.

Saturation optics may additionally include imaging optics and/or phasemodifying optics for coding a wavefront of electromagnetic energytransmitted therethrough, such as described in U.S. Pat. No. 5,748,371,which is included herein by reference. In certain applications, phasemodifying optics may include non-rotationally symmetric optics.Particularly when forming images of distant points with highillumination intensity, phase modifying optics may produce images thatare perceived to be very different from those produced by an imagingsystem including only rotationally symmetric optics (i.e., a“traditional” imaging system without the phase modifying optics).

The use of saturation optics in an imaging system may provide certainbenefits. For example, an imaging system including saturation opticscombined with phase modifying optics may be configured to produce imagesthat appear to have been generated by traditional imaging systems butwith added advantages, such as reduced aberrations and/or extended depthof field. Saturation optics may also be used in imaging systems toproduce images that appear to have been produced by non-traditionalimaging systems with certain identifiable characteristics (e.g., digitalwatermarking) that provide indicators of the optics used in the imagingsystem. An additional consideration is that off-axis saturated imagesformed by a traditional imaging system, when saturated, may exhibitundesirable effects such as, for example, vignetting when an aperturestop is placed against an optical element in the traditional imagingsystem. In practice, the aperture stop has a finite thickness such thatdifferent field positions will overlap in different physical positions,thereby creating an asymmetric saturated response that may reduce theimage quality. The use of saturation optics may thus alleviate suchproblems.

Herein, many of the embodiments are described in association with imagesof points or point objects or, alternatively, PSFs. Within the contextof the present description, such descriptions are consideredinterchangeable. It is further recognized that any object may bedecomposed into a set of point objects and its image may be decomposedinto an associated set of PSFs.

In the context of the present disclosure, a distinction is made betweena PSF in a conventional sense, which is a function of the optics of animaging system, and a “sampled PSF”, which refers to a PSF as capturedby a detector. That is, a sampled PSF is a PSF modified by certaincharacteristics of a detector in the imaging system such as, but notlimited to, sampling patterns, pixilation, wavelength selection andsaturation. The characteristics of a sampled PSF are directly related tothe design of an imaging system as well as the incident electromagneticenergy distribution imaged by the imaging system. Characteristics of asampled PSF include any discernable feature used to describe the shapeand form of a sampled PSF such as, but not limited to contours,outlines, footprints, spatial extents, profiles, cross-sections, pixelvalues, orientation, gradients, intensities and location within animage.

Moreover, a sampled PSF may be further classified as either a saturatedsampled PSF, when the intensity of illumination is sufficient tosaturate a portion of the detector, or an unsaturated sampled PSF, whenthe detector is unsaturated. An associated set of PSFs from an image mayalso be modified by the aforementioned characteristics of a detector inthe imaging system and, when the detector is saturated, result in asampled image with saturated sampled PSF or, more concisely, a“saturated sampled image.”

FIG. 1 is a schematic diagram of an imaging system 100 includingsaturation optics, shown here to illustrate a response of a detector inthe imaging system to non-saturating illumination with a lowillumination level, in accordance with an embodiment. Imaging system 100images a small, off-axis object 120 that is not reflecting or radiatingenough electromagnetic energy to saturate the detector. Imaging system100 also includes saturation optics 140 for imaging object 120 at adetector 160 (e.g., film, CCD, CMOS detector or microbolometer).Saturation optics 140 may include imaging optics and phase modifyingoptics that have been designed to accommodate saturation conditions, asdiscussed in more detail below. Detector 160 and an optional, signalprocessor 170 cooperate to generate electronic data that may be used toform an image 180. If object 120 is sufficiently small, then image 180may be considered a sampled PSF for imaging system 100. Underunsaturated conditions, image 180 is similar to that expected from aconventional imaging system; that is, image 180 includes a shape 120′resembling object 120. Signal processor 170 may further process image180 by application of techniques such as, but not limited to, filtering,scaling and color correction.

FIG. 2 is a schematic diagram of an imaging system 200 with saturationoptics, shown here to illustrate a response of the imaging system tosaturating illumination with a high illumination level. The componentsof imaging system 200 are essentially unchanged from those of imagingsystem 100, with the exception that imaging system 200 is being used toimage a high intensity object 220. The illumination level of highintensity object 220 is large enough so that a saturated sampled image280 results. Saturation optics 140 and detector 160, optionally incombination with signal processor 170, generate electronic datacorresponding to saturated sampled image 280 that may exhibit differentcharacteristics from unsaturated sampled image 180, FIG. 1, or from asaturated sampled image-captured by a traditional imaging system withoutsaturation optics. In the exemplary case shown in FIG. 2, saturatedsampled image 280 is a superposition of a formed image representing text“CDM” and image 180 (note that a dot similar to that in image 180 islocated in the center of the “D” in saturated sampled image 280). Theformed image is produced by saturation optics 140 under saturatedimaging conditions. Such a formed image may be used, for instance, as adigital watermark for use in identification of a given imaging systemunder specific lighting conditions; that is, illumination level may beused to identify a particular imaging system.

Image 280 may be distributed over a large portion of detector 160 andapproximate a sampled PSF formed by a traditional imaging systemincluding, for instance, circularly symmetric optics. As may be seen bycomparing image 180 with saturated sampled image 280, it may be seenthat the use of saturation optics 140 results in identifiably differentsampled images depending only on the illumination intensity of theobject. Saturation optics 140 may also be characterized by a definedfield of view such that the characteristics of a sampled PSFcorresponding to saturated sampled image 280 may depend upon thelocation of object 220 within the field of view. Additionally, the pupilfunction of saturation optics 140 may be further configured such thatimaging an object located outside of the field of view may result in asampled image with still different characteristics. Optionally,saturation optics 140 may be designed such that characteristics of thesampled image, or a portion thereof, is a function of one or more of therange, field angle object, volumetric location, shape, illuminationintensity and color.

A flowchart for a process 300 of designing a pupil function forsaturation optics is shown in FIG. 3. Process 300 may rely upon certaina priori knowledge, such as known diffraction effects at the edge of thepupil, artifacts that may be caused by phase discontinuities andsymmetries to be used in the design of the saturation optics.

Process 300 begins with a START step 310, followed by a step 320 toselect one or more pupil functions as an initial guess for the design.For example, pupil functions 900 of FIG. 9, 1000 of FIG. 10, 3400 ofFIG. 34 and 3500 of FIG. 35 are suitable for selection in step 320. Oncea pupil function has been selected as the initial guess, process 300advances to a step 340, in which the sampled PSF corresponding to theselected pupil function(s) is calculated, taking into accountcharacteristics of the optical elements to be used (e.g., imaging opticsand the selected pupil function) as well as specifications of thedetector to be used with the saturation optics. The sampled PSF may becalculated as a function of such variables as conjugate, saturation andwavelength. The sampled PSF may also be calculated based upon portionsof the pupil function, such as inner and outer regions, rather then theentire pupil function.

Next, in a step 360, the sampled PSF calculated in step 340 is evaluatedby comparison to a predetermined metric. For example, low intensityregions of saturated sampled PSFs and high intensity regions ofunsaturated sampled PSFs may be evaluated to provide characteristics,such as compactness, symmetry, degree of similarity to sampled PSFsproduced by a traditional imaging systems (i.e., without saturatedoptics) and the degree of uniqueness provided in the saturated sampledPSF as compared to the saturated sampled PSF produced by a traditionalimaging system under saturated imaging conditions. Then, a decision 370is made as to whether the sampled PSF evaluated in step 360 isacceptable for the given imaging system. If the answer to decision 370is “YES” the sampled PSF is acceptable, then process 300 ends in a step380. If the answer to decision 370 is “NO” the sampled PSF is notacceptable, then process 300 proceeds to a step 350, at which the pupilfunction is modified in accordance with certain parameters, and theprocess returns to step 340, at which the sampled PSF is calculated forthe pupil function modified in step 350. The pupil function may bemodified, for example, to achieve or maintain one or more of thefollowing properties: constant phase as function of angular coordinate(i.e., Θ in polar coordinates); minimized phase discontinuities asfunctions of radial coordinate (i.e., R in polar coordinates); andcertain characteristics of the associated sampled PSF (e.g., the sampledPSF has circular or closed contours) in both saturated and unsaturatedcases.

One example of a suitable pupil function configuration for saturationoptics is a segmented pupil function. A segmented pupil function mayinclude any number of sectors. For example, FIG. 4 shows a circularpupil function 400 with spatially uniform segmentation into sectors 410,420, 430, 440 and 450. Alternatively, FIG. 5 shows a circular pupilfunction 500 with spatially non-uniform segmentation into six sectors510, 520, 530, 540, 550 and 560 that are further subdivided therein. Thevaried shadings used in FIGS. 4 and 5 are used to represent differentmathematic functional forms. Each sector may have the same or differentfunctional forms, and further include different functional forms and/orshapes within each division of the sector (e.g., sectors 520 and 560,FIG. 5).

One approach to designing a pupil function, including segmentation, isthrough combinations of pupil functions. For example, if P₀ is a firstpupil function that produces a first form of a PSF, P₁ is a second pupilfunction that produces a second form of a PSF and so on, then a newpupil function may be defined as a weighted combination of a variety ofpupil functions:

P _(new) =F(aP ₁ ,bP ₂ , . . . , zP _(z)),  EQN. 1

where a, b and z are parameters. Function F may include any mathematicaloperation such as multiplication, division, addition, division,convolution, nonlinear or combinatorial function or a combinationthereof. Parameters a, b and z may be scalar or vector quantitiesmodifying a pupil function in part or in whole. Examples herein belowdescribe a variety of pupil function designed using this functionalprescription. Saturation optics, including pupil functions defined byEQN. 1, may be combined with signal processing such that that resultingimages of objects and scenes are a function of the intensity of theobject and scenes.

FIG. 6 is an illustration of a circular pupil function 600 withspatially uniform segmentation of first through eighth sectors (610,620, 630, 640, 650, 660, 670 and 680, respectively). Pupil function 600is an example of a constant profile path (“CPP”) form. The CPP form isconstructed using a plurality of straight line segments; that is, eachsector may be mathematically described along straight line segmentsperpendicular to a radial vector from a center of circular pupilfunction 600.

Each sector also has at least two regions defined by the distance fromthe optical axis (e.g., first sector 610 has first and second regions612 and 614, respectively). Regions, such as first region 612, which areclosest to the center of the pupil function are referred to as innerregions, and regions that are farthest from the center of the pupilfunction, such as second region 614, are called outer regions. Forclarity, only first and second regions within first sector 610 have beenlabeled in FIG. 6. In one embodiment, a contour of an outer region of apupil function designed for use with saturation optics may be generallya circular form even though the overall pupil function, especially theinner regions, has large deviation from a circular form. That is,non-circular pupil functions may be designed to generate images ofsaturated objects that appear to have been generated by circular pupilfunctions.

Returning to FIG. 6, the surface shape of each sector varies as afunction of the distance from the center of the pupil function. Themathematical form of a given region (e.g., first region 612) of a sector(e.g., first sector 610) may be expressed by a one-dimensionalmathematical function. For exemplary pupil function 600, all sectors aredescribed by the same polynomial function (e.g., polynomial function 810discussed in reference to FIG. 8). Pupil function 600 is particularlysuited for use in saturation optics due to at least the followingproperties: 1) pupil function 600 has an even-numbered eight-foldsymmetry that produces symmetric PSFs and, consequently, images that areconsidered more “natural” than images produced by PSFs with oddsymmetry; 2) the eight-fold symmetry of pupil function 600 providessufficient modulation and intensity for PSFs in horizontal, vertical andespecially diagonal directions to which the human eye is sensitive; and3) the eight-fold symmetry of pupil function 600 is well suited forintegration with the square lattice and Bayer patterning of digitaldetectors.

FIG. 7 illustrates certain details of one sector of the circular pupilfunction shown in FIG. 6. Like the eight sectors in pupil function 600,a sector 700 includes an inner region 710 and an outer region 720.Mathematical relationships between the clear aperture radius (“CR”) andthe surface form height or sag are given by the following equations:

$\begin{matrix}{{x_{m\; {ax}} = {{CR} \cdot {\cos \left( \frac{\pi}{8} \right)}}},} & {{EQN}.\mspace{14mu} 2} \\{{{{sag}(x)} = {\sum\limits_{n}{\alpha_{n}x^{\beta_{n}}}}},} & {{EQN}.\mspace{14mu} 3} \\{{\forall{x > x_{m\; {ax}}}},{{{sag}(x)} = {{sag}\left( x_{{ma}\; x} \right)}},} & {{EQN}.\mspace{14mu} 4}\end{matrix}$

wherein x is a generalized one-dimensional coordinate, α_(n) is acoefficient and β_(n) is an exponent. Additionally, all sag values alonga chord line 730 are defined to have the same value. While the surfacesag of first region 710 is expressed by a polynomial, the form of outerregion 720 may be determined by other functions.

FIG. 8 shows a plot 800 of an exemplary polynomial (indicated within aninset box 810) defining a pupil function. Polynomial 810 is an exampleof a CPP function. The horizontal axis of plot 800 denotes a normalizedpupil coordinate (i.e., distance from the center of the pupil function)where 0 is the center of the pupil function and 1 is at the edge of thepupil function aperture. The vertical axis denotes height of the surfacesag in units of wavelengths. A CPP function, such as polynomial 810, notonly defines the form of a physical surface of a phase modifying opticalelement or elements but also, when properly scaled, defines amathematical relationship for phase of the electromagnetic energy as afunction of the pupil coordinate.

The mathematical prescriptions associated with FIGS. 6-8 may be combinedto define a pupil function. FIG. 9 shows a 3D mesh plot 900 of anexemplary, pupil function so formed from the combination of theaforedescribed mathematical prescriptions. For mesh plot 900 and othermesh plots herein the x- and y-axes represent arbitrary spatialcoordinates of the pupil functions. Vertical axes of these plotsrepresent surface sag in wavelengths. The pupil function represented bymesh plot 900 is suitable for use in a saturation optics configurationcapable of producing images with, for instance, certain identifiablecharacteristics.

The pupil function represented by mesh plot 900 may be modified using asmoothing function, such as shown in FIG. 10 as a 3D mesh plot 1000.Point-by-point multiplication of the pupil function with the smoothingfunction results in a modified pupil function, such as shown in FIG. 11as a 3D mesh plot 1100. The modified pupil function represented by meshplot 1100 is suitable for use in a saturation optics configuration in anembodiment.

Smoothing functions may be any function that provides desired “damping”characteristics such as, but not limited to, exponential functions,Fermi functions, Einstein functions, Gaussian functions and sigmoidfunctions. The smoothing function represented by 3D mesh plot 1000 is acomplementary error function (“erfc”), which is an example of a sigmoidfunction. To form the erfc smoothing function, such as that representedby mesh plot 1000 of FIG. 10, a one-dimensional function erfc istransformed into to a rotationally symmetric cylindrical form (i.e.,erfc(x)→erfc(r)).

A smoothing function may be selected in regard to its ability to provideat least some of the following advantages: 1) the pupil function becomesconstant at radii beyond a zero slope value of a CPP polynomial (c.f.,in plot 800, f′(x=˜0.76)=0); 2) the pupil function is circularlysymmetric at the pupil function aperture; and 3) the slope of pupilfunction in the radial direction is essentially a constant for all polarangles in the outer region of the pupil function. The smoothing functionmay be designed such that the transition between the inner and outerregions of the modified pupil function occurs in a region of constantslope or at a fixed radius from the center of the pupil function. In anembodiment, the inner regions of the original pupil function and themodified pupil function may remain essentially the same, such as shownin FIG. 11. The outer regions may be highly modified and quitedifferent, as shown by mesh plot 1100.

Alternatively, an apodizing function for modifying the intensity of theelectromagnetic energy may also be used to produce a modifying pupilfunction so as to produce saturation optics that modifies both phase andintensity of electromagnetic energy transmitted therethrough. That is, apupil function for use in a saturation optics configuration may also beformed from only apodizing, intensity modifying functions.

FIGS. 12-15 show plots of sampled PSFs associated with the pupilfunction represented by mesh plot 900 under varying degrees ofsaturation. The x- and y-coordinate units in these plots and subsequentsampled PSF plots herein are in units of detector pixels. FIG. 12 showsan unsaturated sampled PSF 1200, which appears very small and compact.FIG. 13 shows a 10× saturated sampled PSF 1300, which remains small andcompact in appearance although larger than unsaturated sampled PSF 1200.As the illumination intensity is increased to saturation levels of 50×and 500× as shown in FIGS. 14 and 15 respectively, the saturated sampledPSFs become very different from unsaturated sampled PSF 1200. 50× and500× saturated sampled PSFs 1400 and 1500, respectively, have star-likeappearances and are very different in appearance from those formed bytraditional imaging systems. While these effects are often not desirablein consumer applications and may be considered to represent poorlyimaged objects, such effects may be used for specialized purposes suchas watermarking and artistic imaging.

FIGS. 16-19 show plots of sampled PSFs associated with the modifiedpupil function represented by mesh plot 1100 of FIG. 11 under varyingdegrees of saturation. An unsaturated sampled PSF 1600 and a 10×saturated sampled PSF 1700 appear similar to those associated with pupilfunction represented by mesh plot 900. However, the sampled PSFs atgreater levels of saturation (50× and 500× saturated sampled PSFs 1800and 1900, respectively) appear quite different from 50× and 500×saturated sampled PSFs 1400 and 1500. 50× and 500× saturated sampledPSFs 1800 and 1900 appear similar to what a user would expect from thoseassociated with traditional imaging systems. Processing of imagesproduced by saturation optics associated with the PSFs shown in FIGS.12-19 may alter characteristics of the images, for example, a filter maybe used to enhance or at least partially remove the “star like” contoursof saturated sampled PSFs 1400 and 1500 by processing.

FIGS. 20-24 show an exemplary decomposition of a pupil function into aninner region and an outer region. FIG. 20 shows mesh plot 900 of thepupil function of FIG. 9, repeated here for convenience. FIGS. 21 and 22show 3D mesh plots 2100 and 2200, respectively, of an exemplary innerregion masking function and an outer region masking function. A commonradial boundary for the regions has been defined, for this example, tobe at a normalized radius of r=0.77. This normalized radius valueboundary has been chosen such that the slope of the radial CPPpolynomial of FIG. 8 is nearly zero. Each masking function has a valueof one where that portion of the pupil function is to be selected and avalue of zero where the pupil function is not to be selected. FIGS. 23and 24 show 3D mesh plots 2300 and 2400, respectively, of selected innerand outer region pupil functions that result from point-by-pointmultiplication of pupil function shown in FIG. 20 with the maskingfunctions plotted in FIGS. 21 and 22.

FIGS. 25 and 26 show plots of saturated sampled PSFs associated withselected inner region and outer region pupil functions of FIGS. 23 and24, respectively. The x- and y-axis units are in units of detectorpixels. A saturated sampled PSF 2500, associated with the inner regionpupil function of FIG. 25, is as compact as an unsaturated sampled PSF(e.g., unsaturated sampled PSFs 1200 and 1600 of FIGS. 12 and 16,respectively), while a saturated sampled PSF 2600, associated with theouter region pupil function of FIG. 26, exhibits a large star-likeshape. The relative size of saturated sampled PSF 2500 is also muchsmaller than that of saturated sampled PSF 2600. Therefore, it appearsthat the outer region pupil function dominates the effects on the PSF ofthe pupil function as a whole.

FIGS. 27-31 show an exemplary decomposition of the modified pupilfunction, as represented by mesh plot 1100 of FIG. 11, into inner andouter regions. FIG. 27 shows mesh plot 1100 of the modified pupilfunction of FIG. 11, repeated here for convenience. FIGS. 28 and 29 showthe 3D mesh plots 2100 and 2200, respectively, of the masking functionsof FIGS. 21 and 22, also repeated here for convenience. FIGS. 30 and 31show 3D mesh plots 3000 and 3100, respectively, of the selected innerand outer region pupil functions that result from point-by-pointmultiplication of pupil function of FIG. 27 with the masking functionsplotted in FIGS. 28 and 29.

FIGS. 32 and 33 show plots of saturated sampled PSFs associated withselected inner and outer region pupil functions of FIGS. 32 and 33,respectively. A saturated sampled PSF 3200, associated with the innerregion pupil function of FIG. 32, is as compact as an unsaturatedsampled PSF and substantially the same as that of FIG. 25. Saturatedsampled PSF 3300 associated with the outer region pupil function of FIG.31, is quite different from that of FIG. 26. As is apparent in FIG. 33,saturated sampled PSF 3300 associated with the outer region pupilfunction appears circular and closed; that is, saturated sampled PSF3300 appears to be the type of contour that users of traditional imagingsystems have come to expect under saturated imaging conditions.Therefore, tailoring the outer region pupil function appears to have adirect effect on the shape and size of the saturated sampled PSFassociated therewith.

It is recognized herein that the effect of the pupil function underunsaturated and saturated imaging conditions differ significantly.Therefore, the design goals for saturation optics may include, forexample, the ability to: 1) generate suitable forms and proportions ofP₀ and P₁ such that sufficient image quality is provided when the systemis not saturated; and 2) generate suitable forms of P₁ such that whenthe imaging system is saturated, the resulting saturated image is asdesired. In the case of a pupil function with a circular aperture, forinstance, it may be desirable that the slope of the surface around theperiphery of the aperture remain constant as a function of field angleand that there are no phase discontinuities in the radial direction. Fornon-circular apertures, non-constant slopes at the periphery may bedefined so that a saturated sampled image appears similar to that formedby a traditional imaging system. For identification and/or artisticpurposes these design goal may be greatly modified. Such design goalsmay be used as metrics in step 360 of process 300 of FIG. 3 to evaluatethe sampled PSF.

Another method for constructing a composite pupil function is byaddition of two or more full pupil functions. FIGS. 34-36 show anexample of additive construction of a pupil function. FIG. 34 shows acontour plot 3400 of an exemplary pupil function P₁, based upon themathematical form: R³ cos(3Θ), where R is a normalized radial pupilcoordinate and Θ is an angular pupil coordinate. Pupil function P₁ has apeak-to-valley wavefront variation of approximately 1.3 waves.

FIG. 35 shows a grayscale image 3500 of a statistically designed pupilfunction P₂ with a peak to valley wavefront variation of approximately0.35 waves. The actual form of this pupil function is statistically andspatially correlated about the pupil function. The design processinvolves modeling the values across the pupil function as Gaussianrandom variables of unit variance and then convolving the twodimensional (“2D”) random spatial variables with a second order Gaussianfunction of unit volume. The resulting correlated random variables,including amplitude scaling, become the values for pupil function P₂.This type of design process is seen to be generally similar to that usedto design diffractive phase components. For example, bar code scannersmay use such diffractive components to produce alignment marks withprojected illumination systems. Also, novelty business cards may beembossed with simple diffractive optics to produce grayscale images ofkeyboards and general scenes when illuminated by coherentelectromagnetic energy. Techniques to design such diffractive optics areadaptable for use in the design of components for saturation optics.Pupil functions P₁ and P₂ may added to form saturation optics pupilfunction P₁+P₂, represented by a grayscale image 3600 shown in FIG. 36.

FIG. 37 shows a grayscale image of an unsaturated sampled PSF 3700associated with pupil function P₁ of FIG. 34. Unsaturated sampled PSF3700 is generally compact and nearly rotationally symmetric. FIG. 38shows a grayscale image of a saturated sampled PSF 3800 associated withpupil function P₁ of FIG. 34. Saturated sampled PSF 3800 showsnon-rotationally symmetric characteristics. As discussed above,non-rotationally symmetric characteristics may be considered lesspleasing or of lower image quality when compared to a saturated sampledPSF produced by a traditional imaging system. Alternatively, thenon-rotationally symmetric characteristics may be advantageously used asidentification marks.

A grayscale image of an unsaturated sampled PSF 3900 associated withpupil function P₁+P₂ of FIG. 36 is shown in FIG. 39. A comparison ofFIGS. 37 and 39 shows that unsaturated sampled PSFs 3700 and 3900 arevery similar. FIG. 40 shows a grayscale image of a saturated sampled PSF4000 associated with pupil function P₁+P₂ of FIG. 36. A comparison ofFIGS. 38 and 40 shows that saturated sampled PSFs 3800 and 4000 are verydifferent; that is, while saturated PSF 3800 exhibited non-rotationallysymmetric characteristics, saturated PSF 4000 of FIG. 40 is generallycircularly symmetric and is similar to a saturated PSF produced by atraditional imaging system.

The sampled PSFs may be processed by a signal processor, such as signalprocessor 170 of FIG. 1, so as to alter their characteristics. Forexample, a filter may be used to either enhance or at least partiallyremove the three-fold symmetric contours of saturated sampled PSF 3800or a filter may be used to increase the compactness (i.e., decrease thespatial extent) of unsaturated sampled PSFs 3700 and 3900 and saturatedsampled PSF 4000.

Instead of designing pupil function P₂ of FIG. 35 to form aGaussian-like sampled PSF response from the imaging system, the pupilfunction may be tailored to form an image that represents text (such asthe text “CDM” shown in FIG. 2) or, as another example, a model numberof a given imaging system. When such a pupil is interrogated with a highillumination intensity source, such as a laser, the image produced maythen act as an identifier of the type of imaging system. For instance,this effect may be a function of the location of the image on the imageplane by placing function P₂ in a region of imaging system that isilluminated by only off-axis illumination.

Certain changes may be made in the imaging system and processesdescribed herein without departing from the scope thereof. For example,although the pupil functions described herein consider phase only fordesigning saturation optics; intensity or both phase and intensity maybe used. Furthermore, although some of the described embodiments notethe use of two components for forming a saturation optics pupilfunction, more than two may be readily used. It should thus be notedthat the matter contained in the above description or shown in theaccompanying drawings should be interpreted as illustrative and not in alimiting sense. The following claims are intended to cover all genericand specific features described herein, as well as all statements of thescope of the present method and system, which, as a matter of language,might be said to fall there between.

1. An imaging system for imaging electromagnetic energy, comprising: adetector for receiving the electromagnetic energy and generating sampleddata in accordance with the electromagnetic energy so received, thedetector being characterized by a threshold point such that the sampleddata is in one of two states: i) below threshold, when the intensity ofthe electromagnetic energy so received is less than the threshold point;and ii) above threshold, when the intensity of the electromagneticenergy is greater than the threshold point; and saturation optics forproviding a characteristic of the sampled data, wherein a characteristicof the sampled data when below threshold is different from thecharacteristic of the sampled data when above threshold.
 2. The imagingsystem of claim 1, wherein the sampled data comprises a sampled pointspread function (“PSF”).
 3. The imaging system of claim 2, wherein thesampled PSF is saturated when above threshold.
 4. The imaging system ofclaim 3, wherein the sampled PSF exhibits one of a circular contour anda closed contour when above threshold.
 5. The imaging system of claim 3,the imaging system being characterized by a field of view, wherein thesampled PSF, when saturated, exhibits different characteristics for highillumination intensity objects located within the field of view andthose located outside of the field of view.
 6. The imaging system ofclaim 3, the imaging system being characterized by a field of view,wherein the sampled PSF, when saturated, exhibits differentcharacteristics for high illumination intensity objects located atdifferent locations within the field of view.
 7. The imaging system ofclaim 6, the imaging system including stray electromagnetic energypresent therein, further comprising a signal processor for modifying aportion of the sampled data associated with the stray electromagneticenergy in accordance with the sampled PSF.
 8. The imaging system ofclaim 2, further comprising a signal processor for processing thesampled data in accordance with the sampled PSF.
 9. The imaging systemof claim 1, wherein saturation optics comprises imaging optics fordirecting the electromagnetic energy toward the detector and phasemodifying optics for modifying a wavefront of the electromagneticenergy.
 10. The imaging system of claim 9, wherein the phase modifyingoptics are integrally formed with the imaging optics.
 11. The imagingsystem of claim 1, wherein the saturation optics comprise an arrangementof segments providing a multi-fold symmetry.
 12. The imaging system ofclaim 11, wherein each one of the segments is characterized by a surfacesag describable by a one dimensional function along a straight lineperpendicular to a radial vector from a center of the saturation optics.13. The imaging system of claim 12, wherein the one dimensional functioncomprises:${{{sag}(x)} = {\sum\limits_{n}{\alpha_{n\;}x^{\beta_{n}}}}},{where}$$x_{{ma}\; x} = {\left( {{clear}\mspace{14mu} {aperture}\mspace{11mu} {radius}} \right) \cdot {\cos \left( \frac{\pi}{8} \right)}}$and sag(x) = sag(x_(ma x))  for  x > x_(m ax).
 14. The imagingsystem of claim 13, wherein the one dimensional function comprises:sag(x)=−2x+2x ⁹.
 15. The imaging system of claim 11, wherein each of thesegments further comprises at least inner and outer regions havingdifferent surface profiles.
 16. The imaging system of claim 15, whereinthe outer regions of the plurality of segments are uniform and constant.17. The imaging system of claim 15, wherein the outer regions of theplurality of segments are modified with a smoothing function.
 18. Theimaging system of claim 17, wherein the smoothing function comprises asigmoid.
 19. The imaging system of claim 18, wherein the sigmoidcomprises a complementary error function (“erfc”).
 20. The imagingsystem of claim 11, wherein the arrangement comprises segments that areuniform in size.
 21. The imaging system of claim 20, wherein thearrangement comprises eight segments exhibiting an eight-fold symmetry.22. The imaging system of claim 21, wherein the sampled PSF when thedata is above threshold exhibits a predetermined pattern.
 23. Theimaging system of claim 22, wherein the predetermined pattern comprisesone of a star pattern and a digital watermark.
 24. The imaging system ofclaim 23, wherein the digital watermark is viewable by illuminating theimaging system with off-axis and out-of-field illumination.
 25. Theimaging system of claim 11, wherein at least one of the segments of thearrangement is different in size compared to other segments in thearrangement.
 26. Phase modifying optics for use in an imaging system,comprising: a constant profile path form including a plurality ofsectors, each of the plurality of sectors comprising a surface sag(“sag(x)”) describable by a one dimensional mathematical function alonga mid-line corresponding to a radial vector, which radial vector has anorigin x=0 at a center of a circular pupil function and passes through amid-point of a sector chord line at x=x_(max), and a plurality ofstraight line segments perpendicular to the mid-line and having a valueequal to sag(x) therealong, wherein the one dimensional mathematicalfunction comprises:${{{sag}(x)} = {{{- 2}x} + {2x^{9}}}},{{{where}\mspace{14mu} x_{{ma}\; x}} = {{\left( {{clear}\mspace{14mu} {aperture}\mspace{14mu} {radius}} \right) \cdot {\cos \left( \frac{\pi}{8} \right)}}\mspace{14mu} {and}}}$sag(x) = sag(x_(ma x))  for  x > x_(m ax). 27-28. (canceled)29. Phase modifying optics of claim 26, wherein each of the plurality ofsectors further comprises at least inner and outer regions havingdifferent surface profiles.
 30. Phase modifying optics of claim 29wherein the outer regions of the plurality of sectors are uniform andconstant.
 31. Phase modifying optics of claim 26, further comprising amodification of the constant profile path form with a smoothingfunction.
 32. Phase modifying optics of claim 31, wherein the smoothingfunction comprises a sigmoid.
 33. Phase modifying optics of claim 32,wherein the sigmoid comprises a complementary error function (“erfc”).34. Phase modifying optics of claim 26, wherein the plurality of sectorsare same in size.
 35. Phase modifying optics of claim 26, wherein atleast one of the plurality of sectors is different in size compared toother ones of the plurality of segments.
 36. A method for designing apupil function for use in an imaging system, comprising: selecting apupil function; calculating a sampled PSF, taking into accountcharacteristics of the imaging system and the pupil function; evaluatingthe sampled PSF in accordance with a selected metric; and if the sampledPSF does not conform within the selected metric, then modifying thepupil function, using a set of parameter modifications, and repeatingthe evaluating and modifying of the pupil function until the sampled PSFconforms within the selected metric.